{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T00:39:41Z","timestamp":1791851981674,"version":"4.3.4"},"reference-count":29,"publisher":"Institute of Mathematical Statistics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,8,1]]},"abstract":"This paper investigates the theoretical foundation and develops analytical formulas for sample size and power calculations for causal inference with observational data. By analyzing the variance of an inverse probability weighting estimator of the average treatment effect, we decompose the power calculation into three components: propensity score distribution, potential outcome distribution and their correlation. We show that to determine the minimal sample size of an observational study, in addition to the standard inputs in the power calculation of randomized trials, it is sufficient to have two parameters: an overlap coefficient and a confounding coefficient, which quantify the strength of the confounder-treatment and the confounder-outcome association, respectively. For the overlap coefficient, we propose using the Bhattacharyya coefficient, which measures the covariate overlap and, together with the treatment proportion, leads to a uniquely identifiable and easily computable propensity score distribution. For the confounding coefficient, we propose a sensitivity parameter bounded by the R-squared statistic of the regression of the outcome on covariates. Our procedure relies on a parametric propensity score model and a semiparametric restricted mean outcome model, but does not require distributional assumptions on the multivariate covariates. We develop an associated R package PSpower and an online calculator.<\/jats:p>","DOI":"10.1214\/26-aos2649","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T08:22:00Z","timestamp":1787041320000},"page":"2191-2208","source":"Crossref","is-referenced-by-count":1,"title":["Sample size and power calculations for causal inference in observational studies"],"prefix":"10.1214","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0949-7565","authenticated-orcid":false,"given":"Bo","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Statistical Science, Duke University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6759-0725","authenticated-orcid":false,"given":"Chengxin","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Biostatistics and Bioinformatics, Duke University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0390-3673","authenticated-orcid":false,"given":"Fan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Duke University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"4","unstructured":"BILLINGSLEY, P.<\/small> (1995). 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